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Optimal control of a tumor growth model with hyperbolic relaxation of the chemical potential

Pierluigi Colli, Elisabetta Rocca, Jürgen Sprekels

Abstract

In this paper, we study the optimal control of a phase field model for a tumor growth model of Cahn--Hilliard type in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. Both the cases when the double-well potential governing the phase evolution is of either regular or logarithmic type are covered by the analysis. We show the Fréchet differentiability of the associated control-to-state operator in suitable Banach spaces and establish first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. The necessary optimality conditions are then used to derive sparsity results for the optimal controls.

Optimal control of a tumor growth model with hyperbolic relaxation of the chemical potential

Abstract

In this paper, we study the optimal control of a phase field model for a tumor growth model of Cahn--Hilliard type in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. Both the cases when the double-well potential governing the phase evolution is of either regular or logarithmic type are covered by the analysis. We show the Fréchet differentiability of the associated control-to-state operator in suitable Banach spaces and establish first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. The necessary optimality conditions are then used to derive sparsity results for the optimal controls.
Paper Structure (9 sections, 13 theorems, 162 equations)

This paper contains 9 sections, 13 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. General setting and the state system
  3. The case of regular potentials
  4. The linearized system
  5. Existence of optimal controls
  6. Differentiability of the control-to-state operator
  7. First-order necessary optimality conditions
  8. The case of singular potentials
  9. Some remarks on sparsity

Key Result

Theorem 2.2

Suppose that const:weak--h:weak are fulfilled, and let the initial data satisfy Then the state system --ss1 has for every $\,{\bf u}=(u_1, u_2) \in {L^\infty(Q)}\times{L^2(Q)}$ a unique solution $(\mu,\varphi,\sigma)$ in the sense of Definition DEF:WEAK, and there exists a constant $K_1>0$, which depends only on the norm $\,\|{\bf u}\|_{{L^\infty(Q)}\times{L^2(Q)}}$ and the dat Moreover, whenever

Theorems & Definitions (26)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 16 more